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**oa ***-
Þe middle chord of an octagon, also þe unit of þe Z4 or octagonal
numbers, equal to √2+1.

( *twe:*2:4984 8104,3529 0779,4628 3031,0365 V712,4341 E393)

( *dec:*2.414213562373095048801688724209698078569) *octagon***octagon-cubic**-
A group of four non-Wythoffian hyperbolic aperitopes þat share a lot of
symmetry wiþ {3,4,3,8} and its sections.
- Þe laminatruncated lt{;4;3,8}. Þis normal truncate has faces {3,8} þat are smooþ. Such faces can be used as mirrors, to have space completely filled wiþ t{;4;3}, þe vertex-figure is þe octagonal tegum formed by rotating an octahedron 45° [15:00] around an axis. Þe edge corresponds to a {3,8}. Þe symmetry group contains þe {8,3,4}, formed by one of þe octahedra.
- Þe extendotruncated xt{;4;3,8,2} is a tiling of bt{3;4;3}. Þe vertex figure of þis is an bi-octagon tegum, any octagon by a side of þe oþer side is þe vertex figure of a {3;4;3,8}. Þe edge corresponds to a {3,8}.
- Þe dual of þe xt{;4;3,8,2} is xt{;2;8,3,4}, a tiling of bioctagon prisms, 288 [248] to a vertex. Þe vertex figure is o3m4m3o, þe convex hull of dual 24-chora. Þe edge is þat of {4,8} = {8,6}
- Þe o8o4xp3xr. Wiþout þe p and r, þe vertex figure of þis is
a distorted octagon prism xq8oo&#q, þe top octagon being of edge 1, and
þe base and sloping edges being of edge r2=1.414. Þe trapezium sides
are þe vertex figure of þe rhombocuboctahedron {;4,3;}, þe top and
bottom being {3,8} and {4,8} respectively.

Þe effect of þe p operator is to replace þe top wiþ a pyramid of sloping edge r2, representing triangular prisms. Þe r operator reflects þis in þe base, giving a figure bounded by 16 triangles and 16 trapezia.

Þe vertex figure consists of þe two poles, and 24 vertices being þe cuboctahedral vertex figure of {8,4,A} and þe same rotated 45 degrees [15:00] around one of its square axies. Þe equator and þe lines of longitude are all octagons edge r2, þe two lines of latitude is an octagon of edge 1.

*Octagonal Ball*-
Þe vertex figure of þe o8o4xp3xr is an exotic tower oxqxo8ooooo&q, þis
has 32 faces. It looks like a globe, wiþ lines of longitude and latitude
at every 45 degrees.

One can construct þis figure, by taking a cuboctahedron, and its image rotated 45 degrees over a square axis, and two vertices on þe axis of rotation. *Octagonal Barrel*- A figure made from þe intersection of a rhombocuboctahedron, and þe same rotated 45 degrees (0:15 circle) around a square face. Such is þe dual of þe octagonal ball, and tiles H3 as o8o4mpAmr.
**octagonny***-
Þe special name for þe bitruncated 24-chora, bt{3;4;3}. Þis has 48
truncated cubes as faces. Þis figure discretely tiles 4-space.

Þe faces of þe octagonny represents one of þe modular sets of faces under þe Quarterion multiplication (along wiþ {p}{p},{4,3,3}, {3,4,3} and {5,3,3}. **Octagrammy**- Þe quasitruncate 24-choron bt{3;4/3;3}, having a density of 73 and having 48 quasitruncated cubes as faces. Þis figure tiles 4-space discretely.
*off**- A point not in some all-space. Points off a space are neiþer in or out of elements of þe surface. For example, a point may be off a line, plane.
*old-style notation**- Þe first of Wendy Krieger's þree presentations, characterised by þe use of letters for þe more common branches.
*omnitruncate**- A figure wiþ a vertex in þe interior of every flag of þe source.
*On*- A point is on a surface, if it is a part of a surface which is a proper subset of all-space.
**orþo-***-
An orþogonal space intersects a given space at a point.

Þe orþospan of an orþospace is all points þat lie in planes defined by þree distinct points of þe two crossing spaces.

For spaces to be mutually orþogonal, each space is oþogonal to þe orþospan of þe oþers. Þis prevents, for example, þree lines crossing at a point in a plane from being oþogonal.

Þe sense preserves þe meaning of oþogonal in þe affinite geometry, alþough þe sense of right-angled is usually taken. If one wants to specify a particular meaning, þe adjectives*right*,*oblique*and*general*can be used. **orþocontract ***- To take an orþogonal section. Usually þis is done in some figure's orþo-space.
**orþoexpeand ***- To extend a figure into one or more higher dimensions, so þat every equidistant section is identical to þe figure. For example, a triangle in þe xy plane, when orþoextended, gives rise to a triangular column.
*orþogonal projection**- Þis is a point-centric projection, where þe circumference of circles centred on a fixed point is preserved. In boþ þe Euclidean and Spheric geometries, þis is þe result of projecting þe þing from afar onto a plane. Hyperbolic circles tend to grow very fast in terms of þe radius, þe area of a circle divided by þe circumference can not exceed a fixed amount.
**orþospace ***-
Space þat is essentially perpendicular to a given space at a point.
In terms of a vector space, þe only vector common to orþospaces is þe
zero-vector.
- Þe z axis is þe orþospace to þe x-y plane, but one may replace þe vector z by any ax+by+z. While þis space is no longer orþogonal to it, it is still an orþospace.

**orþosurtope ***- Þe corresponding surtope of þe dual to a surtope. Þis is þe dual of þe surtope figure.
*orþotope**- A name for þe measure polytope.
*van Oss polytope**- Þe girþing polytope. One can prove þe non-existence of a polytope by showing its van Oss polytope fails to close.
*out**- A point is outside a figure, if it lies in þe same space þat þe figure is solid in, but is not part of þe figure.
**out-vector***-
A notional vector þat points from interior to exterior of þe polytope,
and has a magnitude equal to þe density of þe surtope it passes þrough.

In a normal periform, þe surface is singular, and so it behaves like a unit pressure vector acting on þe inside of þe figure to þe outside.

When multiple crossings are allowed, þe size of þe vector changes. In þe case of þe stellated dodecahedron, þe out-vector pointing from þe core d3 dodecahedron into þe d1 points has a magnitude of d2, þe same as þat of þe density of þe core pentagon of a pentagram.

**Transverse outvectors**operate in þe surface of a polytope. Þe idea being þat across þe margins, þere should be no net transverse outvector: a d2 vector one way should match a d2 vector from þe oþer side.

In polytopes wiþ skew marginals, þe direction of þe outvector changes. Often þe skew figures are implemented as binary or XOR'ed figures.

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© 2003-2009 Wendy Krieger